Method and device for determining and presenting surface charge and dipole densities on cardiac walls

ABSTRACT

The invention discloses a method, a system, a computer program and a device for determining the surface charge and/or dipole densities on heart walls. Using the foregoing, a table of dipole densities ν(P′, t) and/or a table of surface charge densities ρ(P′, t) of a given heart chamber can be generated.

CROSS REFERENCE TO RELATED APPLICATIONS

The present application is a continuation application of U.S. patentapplication Ser. No. 15/435,763, filed Feb. 17, 2017, which is acontinuation application of U.S. patent application Ser. No. 14/865,435,filed Sep. 25, 2015, now U.S. Pat. No. 9,610,024, which is acontinuation application of U.S. patent application Ser. No. 14/547,258,filed to Nov. 19, 2014, now U.S. Pat. No. 9,167,982, which is acontinuation application of U.S. patent application Ser. No. 14/189,643,filed Feb. 25, 2014, now U.S. Pat. No. 8,918,158, which is acontinuation application of U.S. patent application Ser. No. 13/858,715,filed on Apr. 8, 2013, now U.S. Pat. No. 8,700,119, which is acontinuation application of U.S. patent application Ser. No. 12/376,270,filed on Feb. 3, 2009, now U.S. Pat. No. 8,417,313, which is a 371national stage application of Patent Cooperation Treaty Application No.PCT/CH2007/000380 filed Aug. 3, 2007, entitled METHOD AND DEVICE FORDETERMINING AND PRESENTING SURFACE CHARGE AND DIPOLE DENSITIES ONCARDIAC WALLS, which in turn claims priority to Swiss Patent Application1251/06 filed Aug. 3, 2006, which are incorporated herein by reference.

FIELD OF INVENTION

The invention relates to a method, a system, a computer program and adevice for determining the surface charge and/or dipole densities onheart walls in order to locate the origin(s) of cardiac arrhythmias.

BACKGROUND

For localizing the origin(s) of cardiac arrhythmias it is commonpractice to measure the electric potentials located on the inner surfaceof the heart by electrophysiological means within the patient's heart.For example, for this purpose electrode catheters can be inserted intothe heart and moved around while recording cardiac potentials duringnormal heart rhythm or cardiac arrhythmia. If the arrhythmia has aregular activation sequence, the timing of the electric activationmeasured in voltages at the site of the electrode can be integrated whenmoving the electrode around during the arrhythmia, to create a threedimensional map of the electric activation. By doing this, informationon the localization of the source of arrhythmia(s) and mechanisms, i.e.,reentry circuits, can be diagnosed to initiate or guide treatment(radiofrequency ablation).

This mapping procedure is often aided by computer systems generatingthree dimensional maps of catheter positions by localizing the catheterwith the help of magnetic fields (the so called Carto System) ortransthoracic impedances (by Localisa and NavX). Because all the pointsof such maps are obtained by electrode positions in contact with thecardiac surface, this mapping system is called contact mapping. It hasthe inherent limitation that cardiac activation can only be assessedsimultaneously at the points in contact with the myocardium. Hence, aninstant map of the entire cardiac activation is impossible because theentire heart chamber cannot be contacted without compromising bloodcirculation. An instant mapping of the simultaneous electric activationof the heart chamber, however, might be of advantage in unstablearrhythmias of short duration, rendering the mapping procedures (movingthe electrode around during the arrhythmia) too long. In addition, aninstant map of cardiac electric activation might be of advantage duringirregular arrhythmias or arrhythmias with non-constant activationsequences that render integration of activation times from contactmapping impossible. Finally, instant maps of cardiac activation areprobably also faster and easier obtained, than a contact map generatedby time consuming catheters movements to different areas of the heart inall sorts of cardiac arrhythmias.

The disadvantage of contact mapping can be overcome by “non-contactmapping”, which allows for mapping cardiac activation of a heart chambersimultaneously without contact to the cardiac wall. For this purpose,for instance, a multi electrode array mounted on an inflatable ballooncan be inserted into the heart. The geometry of the heart chamber isobtained either (i) by reconstruction of a contact map, which isobtained from integration of movements with an electrode catheter withinthe heart chamber, or (ii) by importing imaging data from computedtomography or MRI (magnetic resonance imaging).

Once the geometry of the cardiac chamber is outlined in a map theinformation of a simultaneous recording of cardiac farfield potentials(unipoles) by the multi electrode array can be extrapolated to thedesired cardiac map using advanced mathematical methods. Thisnon-contact mapping has the advantage that it provides the entireelectric activation measured by farfield unipolar potentials either insinus rhythm or during arrhythmia without the need for moving anelectrode catheter around the cardiac chamber. This allows for a beat tobeat analysis of cardiac activation and, therefore, unstable, irregularor multifocal arrhythmias can be tracked and treated. However, thedisadvantage of non-contact mapping is that it relies on farfieldpotentials, which do not allow for the same precision in localization ascontact mapping (i.e. measuring local electrograms (potentials) ofcardiac activation by touching the endocardium at the site of interestwith a mapping electrode).

Furthermore, non-contact mapping is more prone to artifact generationand interference from potentials generated by cardiac re-polarizationand adjacent heart chambers (atria/ventricles). These drawbacks can beovercome to a certain extent with several filtering techniques. One theother side, in many cases these drawbacks also render the localizationof cardiac arrhythmias a time-consuming frustrating intervention.

Therefore, the advantages of non-contact mapping, i.e. the instantcardiac activation maps, have to be balanced against the disadvantages,i.e. the decreased spatial resolution due to recording of far fieldsignals, filtering of artifacts, etc.

Finally, another method for the non-invasive localization of cardiacarrhythmias is body surface mapping. In this technique multipleelectrodes are attached to the entire surface of the thorax and theinformation of the cardiac electrograms (surface ECG) is measured involtages integrated to maps of cardiac activation. Complex mathematicalmethods are required in order to determine the electric activation in aheart model, for instance, one obtained from CT or MRI imaging givinginformation on cardiac size and orientation within the thoracic cavity.

The disadvantage of both mapping methods, i.e. contact and non-contacttypes, is the representation of the electric activity of the heart bymeans of potentials, that are the result of a summation of electricactivities of many cardiac cells. The integration of all these localelectric ion charges generated by the cardiac cells provides for thepotentials that are measured by current mapping systems.

Therefore, it is an object of the present invention to provide a method,a system, a program and a device for improving precision, accuracy andspatial resolution of cardiac activation mapping, when compared to priorart systems.

SUMMARY OF INVENTION

It was surprisingly found that the use of surface charge and/or dipoledensities and in particular their distribution in a heart chamber is amuch better indicator of cardiac arrhythmias than electric potentials inthe heart.

In a first aspect, the present invention relates to a method fordetermining a database table of surface charge densities (ρ) of at leastone given heart chamber, the surface charge density informationcomprising a table (data values) ρ(P′, t), wherein:

-   -   i) the position P′=(x′,y′,z′) of a point at the wall of the        heart is defined in x, y, z-coordinates,    -   ii) t is the time of measurement for said surface charge        density, and    -   iii) ρ is the surface charge density at said time t and said        position P′ derived from a measured electric potential from a        given heart chamber,

comprising the following steps:

-   -   a) measuring and/or calculating one or more electric        potential(s) V_(e) in one or more position(s) P at a given time        t, and    -   b) transforming V_(e) into said charge density ρ(P′,t) by using        an algorithm suitable for transforming an electric potential        into surface charge density.

In another aspect, the present invention relates to a method fordetermining a database table of dipole densities ν(P′,t) of at least onegiven heart chamber, the dipole density information comprising a table(data values) ν(P′, t), wherein:

-   -   i) the position P′=(x′,y′,z′) of a point at the wall of the        heart is defined in x, y, z-coordinates,    -   ii) t is the time of measurement for said dipole density, and    -   iii) ν is the dipole density at said time t and said position P′        derived from a measured electric potential V_(e) from a given        heart chamber,

comprising the following steps:

-   -   a) measuring and/or calculating one or more electric        potential(s) V_(e) in one or more positions P at a given time t,        and    -   b) transforming V_(e) into said dipole density ν(P′,t) by using        an algorithm suitable for transforming an electric potential        into surface charge density.

Preferably, the electric potential(s) V_(e) can be determined by contactmapping. Equally preferred the electric potential(s) V_(e) can bedetermined by non-contact mapping.

In one embodiment, the above mentioned algorithm method for transformingsaid V_(e) into surface charge density (ρ) or dipole density (ν) in stepb) above employs the boundary element method (BEM).

The geometry of the probe electrode can be ellipsoidal or spherical.

In one embodiment, the measured potential(s) V_(e) can be transformedinto surface charge densities p using the following equation:

$\begin{matrix}{{V_{e}(P)} = {{- \frac{1}{4\; \pi}}{\int_{S_{e}}{\frac{\rho \left( P^{\prime} \right)}{{P^{\prime} - P}}\; d\; {\sigma \left( P^{\prime} \right)}}}}} & (4)\end{matrix}$

wherein:

-   -   Se=surface of endocardium;    -   P′=integration variable running over the entire cardiac wall;        and    -   P=Position of the measuring electrode.

In another embodiment, the measured potential(s) V_(e) can betransformed into dipole densities ν using the following equation:

$\begin{matrix}{{V_{e}(P)} = {\frac{1}{4\; \pi}{\int_{S_{e}}{{\upsilon \left( P^{\prime} \right)}\frac{\partial}{\partial n_{P^{\prime}}}\frac{1}{{P - P^{\prime}}}\; d\; {\sigma \left( P^{\prime} \right)}}}}} & (5)\end{matrix}$

wherein:

-   -   Se=surface of endocardium;    -   P′=integration variable running over the entire cardiac wall;        and    -   P=Position of the measuring electrode.

According to a further aspect of the present invention, provided is asystem for determining a table of surface charge densities ρ(P′, t) of agiven heart chamber, comprising:

-   -   a) one unit for measuring and recording at least one electric        potential V_(e) at a given position P,    -   b) one a/d-converter for converting the measured electric        potentials into digital data,    -   c) a processor that transforms the digital voltage data into        digital surface charge density data, and    -   d) a memory that stores the at least one electric potential        V_(e) and the transformed digital surface charge density data.

In some embodiments, the measuring and recording unit compriseselectrodes configured to measure an electric potential V_(e) whenbrought into contact with at least one part of the heart chamber.

In some embodiments, the measuring and recording unit compriseselectrodes configured to measure an electric potential V_(e) when not incontact with at least one part of the heart chamber.

The system can also comprise an imaging unit that represents the surfacecharge densities ρ(P′, t) as a 2-dimensional image or time-dependentsequence of images.

The system can comprise an imaging unit that represents the surfacecharge densities ρ(P′, t) as a 3-dimensional image or time-dependentsequence of images.

In accordance with another aspect of the invention, provided is a systemthat generates a table of dipole densities ν(P′, t) of a given heartchamber, comprising:

a) a measuring and recording unit that measures and records data used todetermine at least one electric potential V_(e) at a given position P,

-   -   b) an a/d-converter that converts the at least one electric        potentials V_(e) into digital voltage data,    -   c) a processor that transforms the digital voltage data into        dipole charge density data, and    -   d) a memory that stores the at least one electric potential        V_(e) and the transformed dipole charge density data.

The measuring and recording unit can comprise electrodes configured tomeasure an electric potential V_(e) when brought into contact with atleast one part of the heart chamber.

The measuring and recording unit can comprise electrodes configured tomeasure an electric potential V_(e) when not in contact with at leastone part of the heart chamber.

The system can further comprise an imaging unit that represents thedipole densities ν(P′, t) as a 2-dimensional image or time-dependentsequence of images.

The system can further comprise an imaging unit that represents thedipole densities ν(P′, t) as a 3-dimensional image or time-dependentsequence of images.

The system can be configured to implement the above cited methods of theinvention.

In a further aspect, the present invention is directed to a computerprogram comprising instructions for implementing a method of the presentinvention.

In a further aspect, the computer program of the invention can compriseinstructions implementing a system of the invention.

The computer program of the present invention can comprise a computerreadable program code executable by a processor, where the method caninclude starting program after booting a computer and/or a system inaccordance with the invention.

A further aspect of the invention relates to a device for implementing amethod according to the invention, comprising at least one an electrodefor measuring the electrode potential V_(e) using the method of contactmapping and/or using the method of non-contact mapping, at least oneprocessing unit for generating and transforming V_(e) into said surfacecharge density ρ(P′, t) and/or dipole density ν(P′, t) for presenting ona display.

DRAWINGS

FIG. 1 is an exemplary embodiment of a mapping system, according toaspect of the present invention;

FIG. 2 is an exemplary embodiment of a computer architecture formingpart of the mapping system of FIG. 1;

FIG. 3 is an example embodiment of a method of determining and storingsurface charge densities, in accordance with aspects of the presentinvention; and

FIG. 4 is an example embodiment of a method of determining and storingdipole densities, in accordance with aspects of the present invention.

DETAILED DESCRIPTION OF ILLUSTRATIVE EMBODIMENTS

Research has indicated that the use of the surface charge densities(i.e. their distribution) or dipole densities (i.e. their distribution)to generate distribution map(s) will lead to a more detailed and preciseinformation on electric ionic activity of local cardiac cells thanpotentials. Surface charge density or dipole densities represent aprecise and sharp information of the electric activity with a goodspatial resolution, whereas potentials resulting from integration ofcharge densities provide only a diffuse picture of electric activity.The electric nature of cardiac cell membranes comprising ionic chargesof proteins and soluble ions can be precisely described by surfacecharge and dipole densities. The surface charge densities or dipoledensities cannot be directly measured in the heart, but instead must bemathematically and accurately calculated starting from measuredpotentials. In other words, the information of voltage maps obtained bycurrent mapping systems can be greatly refined when calculating surfacecharge densities or dipole densities from these.

The surface charge density means surface charge (Coulombs) per unit area(cm²). A dipole as such is a neutral element, wherein a part comprises apositive charge and the other part comprises the same but negativecharge. A dipole might represent the electric nature of cellularmembranes better, because in biological environment ion charges are notmacroscopically separated.

In order to generate a map of surface charge densities (surface chargedensity distribution) according to the present invention, the geometryof the given heart chamber must be known. The 3D geometry of the cardiacchamber is typically assessed by currently available and common mappingsystems (so-called locator systems) or, alternatively, by integratinganatomical data from CT/MRI scans. FIG. 1 shows an example embodiment ofa mapping system 100 that can be used to map a heart 12 of a human 10.Mapping system 100 can include a computer 110 having known types ofinput devices and output devices, such as a display 120 and printer 130,and a probe system 140. For the measurement of potentials thenon-contact mapping method a probe electrode 142 will be used, which isconnected to the computer 110 via a cable and forms part of probe system140. The probe electrode 142 may be a multi-electrode array withelliptic or spherical shape. The spherical shape has certain advantagesfor the subsequent data analysis. But also other types or even severalindependent electrodes could be used to measure V_(e). For example, whenconsidering, for example, the ventricular cavity within the endocardiumand taking a probe electrode with a surface S_(P), which is located inthe blood, it is possible to measure the potential V(x,y,z) at pointx,y,z on the surface S_(P). In order to calculate the potential at theendocardial surface S_(e) the Laplace equation:

$\begin{matrix}{{\Delta \; V} = {{\left( {\frac{\partial^{2}}{\partial x^{2}} + \frac{\partial^{2}}{\partial y^{2}} + \frac{\partial^{2}}{\partial z^{2}}} \right)V} = 0}} & (1)\end{matrix}$

needs to be solved, wherein V is the potential and x,y,z denote thethree dimensional coordinates. The boundary conditions for this equationare V(x,y,z)=V_(P)(x,y,z) on S_(P), wherein V_(P) is the potential onsurface of the probe.

The solution is an integral that allows for calculating the potentialV(x′y′z′) at any point x′y′z′ in the whole volume of the heart chamberthat is filled with blood. For calculating said integral numerically adiscretisation of the cardiac surface is necessary and the so calledboundary element method (BEM) has to be used.

The boundary element method is a numerical computational method forsolving linear integral equations (i.e. in surface integral form). Themethod is applied in many areas of engineering and science includingfluid mechanics, acoustics, electromagnetics, and fracture mechanics.

The boundary element method is often more efficient than other methods,including the finite element method. Boundary element formulationstypically give rise to fully populated matrices after discretisation.This means, that the storage requirements and computational time willtend to grow according to the square of the problem size. By contrast,finite element matrices are typically banded (elements are only locallyconnected) and the storage requirements for the system matricestypically grow quite linearly with the problem size.

With the above in mind, all potentials V_(P) (x1′,y1′,z1′) on thesurface of the probe can be measured. To calculate the potential V_(e)on the wall of the heart chamber, the known geometry of the surface ofthe heart chamber must be divided in discrete parts to use the boundaryelement method. The endocardial potentials V_(e) are then given by alinear matrix transformation T from the probe potentials V_(P):V_(e)=TV_(P).

After measuring and calculating one or more electric potential(s) V_(e)of cardiac cells in one or more position(s) P(x,y,z) of the at least onegiven heart chamber at a given time t. The surface charge density andthe dipole density is related to potential according to the followingtwo Poisson equations:

$\begin{matrix}{{\Delta \; V_{e}} = {{\rho (P)}{\delta_{S_{e}}(P)}}} & (2) \\{{\Delta \; V_{e}} = {\frac{\delta}{\partial n}\left( {\upsilon \; {\delta_{S_{e}}(P)}} \right)}} & (3)\end{matrix}$

wherein ρ(P) is the surface charge density in position P=x,y,z, δ_(S)_(e) (P) is the delta-distribution concentrated on the surface of theheart chamber S_(e) and ν is the dipole density.

There is a well known relationship between the potential V_(e) on thesurface of the wall of the heart chamber and the surface charge (4) ordipole densities (5).

$\begin{matrix}{{V_{e}(P)} = {{- \frac{1}{4\; \pi}}{\int_{S_{e}}{\frac{\rho \left( P^{\prime} \right)}{{P^{\prime} - P}}d\; {\sigma \left( P^{\prime} \right)}}}}} & (4) \\{{V_{e}(P)} = {\frac{1}{4\; \pi}{\int_{S_{e}}{{\upsilon \left( P^{\prime} \right)}\frac{\partial}{\partial n_{P^{\prime}}}\frac{1}{{P - P^{\prime}}}\; d\; {\sigma \left( P^{\prime} \right)}}}}} & (5)\end{matrix}$

(For a review see Jackson JD. Classical Electrodynamics, 2^(nd) edition,Wiley, New York 1975.)

The boundary element method again provides a code for transforming thepotential V_(e) in formulas 4 and 5 into the desired surface chargedensities and dipole densities, which can be recorded in the database.

In another embodiment of the method of the present invention theelectric potential(s) V_(e) is (are) determined by contact mapping. Inthis case the steps for calculating the electric potential V_(e) are notnecessary, because the direct contact of the electrode to the wall ofthe heart chamber already provides the electric potential V_(e).

In a preferred embodiment of the method of the present invention theprobe electrode comprises a shape that allows for calculating preciselythe electric potential V_(e) and, thus, simplifies the calculations fortransforming V_(e) into the desired charge or dipole densities. Thispreferred geometry of the electrode is essentially ellipsoidal orspherical.

In order to employ the method for determining a database table ofsurface charge densities of at least one given heart chamber in thecontext of the present invention, it is preferred to use a systemcomprising at least:

-   -   a) one unit for measuring and recording electric potentials V at        a given position P(x,y,z) on the surface of a given heart        chamber (Contact mapping) or a probe electrode positioned within        the heart, but without direct wall contact (noncontact mapping)    -   b) one a/d-converter for converting the measured electric        potentials into digital data,    -   c) one memory to save the measured and/or transformed data, and    -   d) one processor unit for transforming the digital data into        digital surface charge density or dipole density data.

It is noted that numerous devices for localising and determiningelectric potentials of cardiac cells in a given heart chamber byinvasive and non-invasive methods are well known in the art and havebeen employed by medical practitioners over many years. Hence, themethod, system, and devices of the present invention do not require anyparticular new electrodes for implementing the best mode for practicingthe present invention. Instead, the invention provides a new andadvantageous processing of the available data that will allow for anincrease in precision, accuracy and spatial resolution of cardiacactivation mapping when compared to prior art systems based on electricsurface potentials in the heart only. In the near future, the presentinvention will allow for providing superior diagnostic means fordiagnosing cardiac arrhythmias and electric status of heart cellsincluding metabolic and functional information.

FIG. 2 provides an example embodiment of a computer architecture 200that can form part of mapping system 100. Architecture 200 includesstandard interface modules 210 for probe system 140 (and electrode 142)and standard interface modules 220 for interfacing with output devices120, 130. The computer includes at least one processor 240 and at leastone computer memory 250. The foregoing are generally known, however thepresent invention further includes an electrical potential to surfacecharge density and/or dipole density converter module 230. Module 230includes instructions necessary for carrying out the methods describedherein, when executed by processor 240, wherein the results of suchprocessing are stored in memory 250—as would be understood by oneskilled in the art having the benefit of this disclosure.

FIG. 3 and FIG. 4 summarize methods for determining and storing surfacecharge densities and dipole densities in accordance with aspects of thepresent invention, respectively, which have been described in detailabove.

In method 300 of FIG. 3, in step 302, mapping system 100 is used tomeasure and/or calculate one or more electric potential(s) V_(e) intoone or more position(s) P within a heart chamber at a given time t. Instep 304, V_(e) is transformed into a surface charge density ρ(P′,t). Instep 306, the surface charge density ρ(P′,t) is stored in a databasetable. The method is repeated if there is another P, in step 308.

In method 400 of FIG. 4, in step 402, mapping system 100 is used tomeasure and/or calculate one or more electric potential(s) V_(e) in oneor more position(s) P within a heart chamber at a given time t. In step404, V_(e) is transformed into said dipole density ν(P′,t) by using analgorithm suitable for transforming an electric potential into surfacecharge density. In step 406, the dipole density ν(P′,t) is stored in adatabase table. The method is repeated if there is another P, in step408.

While the foregoing has described what are considered to be the bestmode and/or other preferred embodiments, it is understood that variousmodifications may be made therein and that the invention or inventionsmay be implemented in various forms and embodiments, and that they maybe applied in numerous applications, only some of which have beendescribed herein. It is intended by the following claims to claim thatwhich is literally described and all equivalents thereto, including allmodifications and variations that fall within the scope of each claim.

What is claimed is:
 1. A method for generating a map of charge densitiesacross an endocardium of a heart chamber, the method comprising: (i)recording a first set of electric potentials by at least one electrodein contact with a surface of the heart; (ii) recording a second set ofelectric potentials by at least one electrode not in contact with asurface of the heart; (iii) transforming the electric potentials into acontinuum of cellular membrane dipole density data and/or surface chargedata using a special purpose computer; and (iv) generating a display ofthe continuum.
 2. The method of claim 1, including performing therecording in step (i) using a plurality of electrodes that includeselectrodes arranged in an ellipsoidal geometry.
 3. The method of claim1, including performing the recording in step (i) using a plurality ofelectrodes that includes electrodes arranged in a spherical geometry. 4.The method of claim 1, including performing the recording in step (ii)using one or more skin electrodes on the thorax.
 5. The method of claim1, including performing step (iii) using a processor executing a set oftransformation instructions stored in a computer memory.
 6. The methodof claim 5, wherein the transformation instructions include rules toemploy a boundary element method (BEM) in transforming the electricpotential data into cellular membrane dipole density in step (iii). 7.The method of claim 6, including employing the BEM for performing adiscretisation of the endocardium.
 8. The method of claim 6, wherein thetransformation instructions implement rules for transforming measuredpotentials V_(e) into surface charge densities p according to thefollowing equation:${V_{e}(P)} = {{- \frac{1}{4\; \pi}}{\int_{S_{e}}{\frac{\rho \left( P^{\prime} \right)}{{P^{\prime} - P}}d\; {\sigma \left( P^{\prime} \right)}}}}$wherein: Se=surface of endocardium; P′=integration variable running overthe entire cardiac wall; and P=Position of the measuring electrode. 9.The method of claim 6, wherein the transformation instructions implementrules for transforming measured potentials V_(e) into dipole densities νaccording to the following equation:${V_{e}(P)} = {\frac{1}{4\; \pi}{\int_{S_{e}}{{\upsilon \left( P^{\prime} \right)}\frac{\partial}{\partial n_{P^{\prime}}}\frac{1}{{P - P^{\prime}}}\; d\; {\sigma \left( P^{\prime} \right)}}}}$wherein: Se=surface of endocardium; P′=integration variable running overthe entire cardiac wall; and P=Position of the measuring electrode. 10.The method of claim 1, including performing step (iv) using a processorexecuting a set of map generation instructions stored in a computermemory.
 11. The method of claim 1, wherein generating the display of thecontinuum in step (iv) includes generating a graphical representation ofthe cellular membrane dipole density data and/or surface charge data inassociation with a graphical representation of the heart.
 12. The methodof claim 1, wherein generating a display of the continuum in step (iv)includes generating a display of a map of dipole densities and/orsurface charge densities as a 2-dimensional image.
 13. The method ofclaim 1, wherein generating a display of the continuum in step (iv)includes generating a display of a map of dipole densities and/orsurface charge densities as a 3-dimensional image.
 14. The method ofclaim 1, wherein generating a display of the continuum in step (iv)includes generating a display of a map of dipole densities and/orsurface charge densities as a time-dependent sequence of images.
 15. Themethod of claim 1, wherein the continuum is a continuum of cellularmembrane dipole density data.
 16. The method of claim 1, wherein thecontinuum is a continuum of cellular membrane surface charge data. 17.The method of claim 1, wherein transforming the electric potentials intothe continuum of cellular membrane dipole density data and/or surfacecharge density data in step (iii) and generating a display of thecontinuum in step (iv) includes determining cellular membrane dipoledensity data and/or surface charge density data at a set of positions P′as a 3-dimensional image, a 2-dimensional image, or a time-dependentsequence of images, or a combination of one or more thereof.
 18. Themethod of claim 1, wherein the recording in steps (i) and step (ii)includes sequentially contact and non-contact by the at least oneelectrode in contact with a surface of the heart and the at least oneelectrode not in contact with a surface of the heart.
 19. The method ofclaim 1, wherein the recording in steps (i) and step (ii) includessimultaneous contact and non-contact by the at least one electrode incontact with a surface of the heart and the at least one electrode notin contact with a surface of the heart.
 20. The method of claim 1,including performing steps (i) and (ii) with a probe system forrecording electric potentials at given positions P on a cellularmembrane of an endocardial wall in a heart chamber to produce theelectric potentials.